Tainted Veins (or the appearance thereof) are a comon consequence of venous thrombosis - a blood clot lodging in a vein and obstructing flow proximal or "upstream" (i.e. back to the heart) from that point on. With the main vein blocked off, smaller collateral veins frequently become engorged with backed-up blood trying to find an alternate path around the blockage. As a result, all the veins distal or "downstream" of the clot will swell (sometimes to grotesque proportions), and those that were close to the skin to begin with will become easily visible on the skin surface as dark purple or blue lines.
A slightly nastier version of the same phenomenon occurs in the context of central retinal vein thrombosis - a blood clot blocking off the main vein draining blood out of the eye. The overall appearance is of an entire eyeball (retina, globe and all) with a case of bloodshot turned Up to Eleven. Cue Eye Scream and/or Squick as appropriate.
In a crossover between Tainted Veins, Blood from the Mouth and High-Pressure Blood, consider portal hypertension. In chronic alcoholism and a few other conditions of massive liver damage, the liver becomes so scarred that blood vessels can't get through it anymore, and as a result the entire blood supply starts backing up, causing high blood pressure in the very fragile veins of the upper GI tract (esophagus and upper stomach mostly). The visible manifestation of this situation, called "caput medusae" ("Medusa's Head"), presents as a dark, snakelike pattern of veins across a swollen abdominal surface. The deeper cousins of these swollen veins, called "esophageal varices," can easily rip open, spilling the patient's entire blood volume into his/her GI tract in a matter of minutes. If you're the physician or nurse dealing with this situation, You May Panic Now.
While not strictly vascular, gangrene can cause reddish or greenish streaks running away from the site.
For a while scientists were experimenting with using Gallium as a substitute iron supplement. Iron is the most critical component of hemoglobin, the chemical compound that allows your blood to carry oxygen. Unfortunately, your body does not absorb most of the iron it takes in, meaning iron supplements have to be taken in large doses (which is bad considering they cause constipation). Gallium is chemically very similar to iron and can from apparently fully functional hemoglobin, however for some reason the human body absorbs nearly 100% of the gallium it takes in. Gallium also has a melting point of about eighty degrees Fahrenheit and will liquefy readily in the human body, meaning it can be ingested while elementally pure, removing the need for any other chemically and their side affects to be added. Unfortunately Gallium also oxidizes blue instead of red, Which gave many of the test animals bright blue visible arteries and veins.